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She lost her temper over the most trivial thing.
Literal
She [topic-は] extremely trivial thing [over-で] snapped.
ごく ('extremely, utterly') intensifies the smallness of the trigger — ごくささいなこと is 'a really tiny thing.' ささい ('trifling, petty') is a classic written-leaning na-adjective; 些細 in kanji is occasional but usually written in kana. The で marks the cause or trigger of the emotional reaction. カッとなる ('to snap, fly off the handle') uses the 擬態語 カッと, miming the sudden mental flare-up — written in katakana, as is conventional for mimetics. The construction ~となる ('become / take on the state of X') after a mimetic is a productive way to verb-ify the snap into emotional explosion.